In order to see an example of such possible fine tuning, consider the Kaon system and observables such as the \(K^0-\bar{K}^0\)  mass difference, or the CP-violating phase \(\epsilon _K\) . The matrix element describing the transition \(\bar{K}^0 \rightarrow K^0\) , \(M_{21}\) , receives contributions from the SM, via box diagrams, and from NP, through tree-level FCNCs in the scalar sector:
\(M_{21} = M_{21}^\mathrm {SM} + M_{21}^\mathrm {NP}\) . The NP terms arise in our model from tree-level Feynman diagrams and thus can in principle overwhelm the SM result. These diagrams represent the tree-level exchanges of CP-even and CP-odd scalars with FCNC interactions which, using the vacuum-insertion approximation
(see Refs. [1]}, [2]}, [3]}), are found to be
\(M_{21}^\mathrm {K,NP}&=& \frac{f_K^2 m_K}{96 v^2} \left\lbrace \frac{10 m_K^2}{\left( m_s + m_d \right)^2}\left( \frac{F^{dA_1}_{ds}}{m_{A_1}^2} + \frac{F^{dA_2}_{ds}}{m_{A_2}^2}- \frac{F^{dh}_{ds}}{m_h^2}- \frac{F^{dH_1}_{ds}}{m_{H_1}^2} - \frac{F^{dH_2}_{ds}}{m_{H_2}^2}\right)\right. \nonumber \\& & \left.+ 4 \left[ 1 + \frac{6 m_K^2}{\left( m_s + m_d \right)^2} \right]\left( \frac{\bar{F}^{d A_1}_{ds}}{m_{A_1}^2} + \frac{\bar{F}^{d A_2}_{ds}}{m_{A_2}^2}+ \frac{\bar{F}^{d h}_{ds}}{m_h^2}+ \frac{\bar{F}^{d H_1}_{ds}}{m_{H_1}^2} + \frac{\bar{F}^{d H_2}_{ds}}{m_{H_2}^2}\right)\right\rbrace  \,,\) 
